The exact value of sec(π/4) is √2.
Understanding sec(π/4)
The secant function (sec) is one of the fundamental trigonometric ratios. It is defined as the reciprocal of the cosine function. Therefore, for any angle x:
sec(x) = 1 / cos(x)
The angle π/4 radians is equivalent to 45 degrees (45°). This specific angle is a common reference angle in trigonometry, often found in a 45-45-90 right-angled triangle or on the unit circle.
Deriving the Exact Value of sec(π/4)
To find the value of sec(π/4), we first need to determine the value of cos(π/4):
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Step 1: Identify the cosine of the angle.
The cosine of π/4 radians (or 45°) is a well-known trigonometric value. In a 45-45-90 right triangle, where the two non-hypotenuse sides are equal, the cosine of 45° is the ratio of the adjacent side to the hypotenuse, which is 1/√2.
cos(π/4) = cos(45°) = 1/√2 -
Step 2: Apply the secant definition.
Since secant is the reciprocal of cosine, we take the reciprocal of 1/√2:
sec(π/4) = 1 / cos(π/4)
sec(π/4) = 1 / (1/√2)
sec(π/4) = √2
This derivation shows that the exact value of sec(π/4) is √2.
Key Equivalents and Values
It's useful to see the relationship between the angle in radians, degrees, its cosine, and its secant value.
| Trigonometric Function | Angle (Radians) | Angle (Degrees) | Exact Value | Decimal Approximation |
|---|---|---|---|---|
| cos(x) | π/4 | 45° | 1/√2 | ≈ 0.70710678 |
| sec(x) | π/4 | 45° | √2 | ≈ 1.414213562 |
As observed, the value of sec(π/4) in decimal form is approximately 1.414213562, but the exact answer, which avoids any rounding, is √2.
Importance of Exact Values
Using exact values like √2 instead of their decimal approximations is crucial in many scientific and engineering calculations. It ensures:
- Precision: No loss of accuracy due to rounding errors, which can compound in multi-step calculations.
- Clarity: Exact values often reveal underlying mathematical relationships more clearly than decimal forms.
- Analytical Solutions: Essential for algebraic manipulation and theoretical analysis in fields such as physics, advanced calculus, and geometry.
For further exploration of trigonometric values, you can refer to resources like Cuemath.
